arXiv subjects

William Hall

Publications and source records attributed to William Hall.

6 recordsLinked to original sources

arXiv2026

Data-adaptive gene and pathway-based tests forrare-variant associations with survival outcomes

Statistical methods for testing aggregate rare-variant genetic associations are typically based on either burden or dispersion tests (or a combination of the two). These methods lack statistical power in the presence of diverse genetic architectures. Moreover, few aggregate rare-variant association methods have been developed specifically for survival data. To address these issues, we propose data-adaptive gene- and pathway-based association tests based on Schoenfeld residuals in Cox proportional hazards models for association studies between an aggregate of rare-variants and survival outcomes. Our methods improve statistical power while maintaining flexibility across various genetic effect sizes and directions. We develop an efficient R package that enables fast computation and supports data simulation as well as gene- and pathway-level testing. Applying our approach to late bladder toxicity following radiotherapy for non-metastatic prostate cancer, we identify biologically relevant genes and pathways, replicate known signals, and capture additional associations. Our method provides a powerful, adaptive framework for survival-based genetic association studies of rare-variants. Keywords: aSPU, time-to-event outcomes, rare-variant associations, Cox regression, Schoenfeld residuals

stat.ME↗

arXiv2007

Numerical evidence for the maximum number of mutually unbiased bases in dimension six

The question of determining the maximal number of mutually unbiased bases in dimension six has received much attention since their introduction to quantum information theory, but a definitive answer has still not been found. In this paper we move away from the traditional analytic approach and use a numerical approach to attempt to determine this number. We numerically minimise a non-negative function of a set of N+1 orthonormal bases in dimension d which only evaluates to zero if the bases are mutually unbiased. As a result we find strong evidence that (as has been conjectured elsewhere) there are no more than three mutually unbiased bases in dimension six.

quant-ph↗

arXiv2007

Compatibility of subsystem states and convex geometry

The subsystem compatibility problem, which concerns the question of whether a set of subsystem states are compatible with a state of the entire system, has received much study. Here we attack the problem from a new angle, utilising the ideas of convexity that have been successfully employed against the separability problem. Analogously to an entanglement witness, we introduce the idea of a compatibility witness, and prove a number of properties about these objects. We show that the subsystem compatibility problem can be solved numerically and efficiently using semidefinite programming, and that the numerical results from this solution can be used to extract exact analytic results, an idea which we use to disprove a conjecture about the subsystem problem made by Butterley et al. [Found. Phys. 36 83 (2006)]. Finally, we consider how the ideas can be used to tackle some important variants of the compatibility problem; in particular, the case of identical particles (known as N-representability in the case of fermions) is considered.

quant-ph↗

arXiv2006

Constructions of indecomposable positive maps based on a new criterion for indecomposability

We give a criterion for a positive mapping on the space of operators on a Hilbert space to be indecomposable. We show that this criterion can be applied to two families of positive maps. These families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states.

quant-ph↗

arXiv2006

Cluster state quantum computation for many-level systems

The cluster state model for quantum computation [Phys. Rev. Lett. 86, 5188] outlines a scheme that allows one to use measurement on a large set of entangled quantum systems in what is known as a cluster state to undertake quantum computations. The model itself and many works dedicated to it involve using entangled qubits. In this paper we consider the issue of using entangled qudits instead. We present a complete framework for cluster state quantum computation using qudits, which not only contains the features of the original qubit model but also contains the new idea of adaptive computation: via a change in the classical computation that helps to correct the errors that are inherent in the model, the implemented quantum computation can be changed. This feature arises through the extra degrees of freedom that appear when using qudits. Finally, for prime dimensions, we give a very explicit description of the model, making use of mutually unbiased bases.

quant-ph↗

arXiv2005

Multipartite reduction criteria for separability

The reduction criterion is a well known necessary condition for separable states, and states violating this condition are entangled and also 1-distillable. In this paper we introduce a new set of necessary conditions for separability of multipartite states, obtained from a set of positive but not completely positive maps. These conditions can be thought of as generalisations of the reduction criterion to multipartite systems. We use tripartite Werner states as an example to investigate the entanglement detecting powers of some of these new conditions, and we also look at what these conditions mean in terms of distillation. Finally, we show that these maps can be used to give a partial solution to the subsystem problem, as described in Ref. [14].

quant-ph↗